Geometric structures of micropolar continuum with elastic and plastic deformations based on generalized Finsler space

Abstract

Elasto-plastic deformations of micropolar continuum are discussed by a non-Riemannian geometry. The non-locality of micropolar continuum is described in a second-order vector bundle of displacements and microrotations. With a decomposition of total elasto-plastic field, geometric quantities are divided into the elastic and plastic components independently. Especially, when an intrinsic parallelism of displacements and microrotations holds, integrability conditions of the elasto-plastic field are represented by a torsion tensor or the curvature of nonlinear connection. Then, Burgers and Frank vectors and an energy release rate around crack tips are related to the torsion tensor or the curvature of nonlinear connection. Moreover, the non-locality of microrotation is discussed based on a kink band as a disclination. It is found a generalized expression of Burgers vector which can describe the kink interface including the disclination.

Keywords

Finsler geometry micropolar continuum elasto-plastic deformation incompatibility condition kink band

1. Introduction

A relationship between topological quantities and geometric objects has been discussed widely in physics. In the continuum theory of defects, a dislocation is regarded as a topological charge created through a condensation of bosons with spontaneous breakdown of symmetries [1, 2]. Such dislocation is geometrically expressed by a torsion tensor [3 –10]. Moreover, an energy integral called J-integral around a disclination field is described by a curvature tensor [11].

From a viewpoint of multi-valued field theory, an existence of the topological singularity is given by an integrability condition of the multi-valued function [12 –14]. Especially, in the quantum field theory, the topological singularity is represented as a macroscopic object called an extended object [1, 2, 15]. A grain boundary and point defect within a crystal is considered as the macroscopic object [2, 16 –20]. Moreover, the properties of the topological singularity in the superconductivity [21], soliton [22], relativistic field theory [23], and neurodynamics [24 –26] have been studied.

The geometric objects such as a curvature tensor and a torsion tensor are related to the topological singularity in the framework of Finsler geometry [27, 28]. Especially, the dislocations and the disclinations are regarded as the singularity in the micropolar continuum (e.g., [29]). Such a generalized continuum with microscopic structure has been studied based on the Finsler geometry which can describe a fluctuation, non-locality, or anisotropy [30 –34]. In this case, the internal state vector is attached to each point of the body, and the finite deformation of nonlinear elastic bodies and the microscopic structure of continuum [35 –40] has been studied. For example, a shear failure is associated with a slipped displacement as the internal state vector [37]. Moreover, boundary value problems on a grain boundary, twin boundaries, and failure interfaces have been studied [41, 42].

From the viewpoint of vector bundle, a hierarchical structure of the microscopic continuum is described by the non-Riemannian geometric quantities. In this case, the local coordinates of bundle are expressed by the internal variables (a line element or an element of support). If the internal state vector [35 –40] is attached to each point of the continuum, then the discussion is on a first-order vector bundle. Especially, in a description of finite deformation of continuum, the line element on the first-order vector bundle is given by a material coordinate and an internal state variable. Then, the curvature tensor and the torsion tensor are obtained from the connection structure, and the geometric approach of the vector bundle can be applied even when the continuum is finitely deformed [28]. Moreover, in generalized continuum such as the Cosserat or micropolar continuum which has often been applied to geophysical field [43 –45, 47, 48] and modeling slip, kink, and shear banding [49], the displacement vector is considered as a non-local variable of the line element. In this case, a first-order vector bundle is used to describe the deformation quantities. Then, the dislocation or the Burgers vector can be represented by the torsion tensor as the topological singularity due to the multi-valued displacement vector [27].

As the number of internal variables increases, the physical field is more non-localized, and the line element is associated with a higher-order vector bundle [50]. When a continuum with microelements is discussed, the microrotations appear that are different from the local rotations of the continuum [51 –54]. For such a more non-local field with microrotation, the deformation of the continuum is described by a second-order vector bundle including the displacement vector and the microrotation vector in the line element. In this case, two types of topological singularities, dislocations and disclinations, can be represented by the geometric quantities [28]. However, the geometric approaches to the micropolar continuum [27, 28] assume only elastic deformation and have not discussed plastic deformation. The elasto-plastic deformations of micropolar continuum have been studied extensively [55 –57], and the deformation quantities are represented by elastic and plastic components that are independent of each other. Moreover, the compatibility from the geometric point of view has been discussed in case of the Cosserat continuum [58]. Such compatibility conditions of the displacement vector and the microrotation vector can also be expressed by the elastic component or the plastic component of the deformation quantities. Therefore, by generalizing the geometrical approaches to the micropolar continuum containing only the elastic component, it is expected to discuss geometric and topological structures of the deformation of the micropolar continuum with the elasto-plastic component.

In this study, we discuss a relationship between the geometric object and the topological singularity in the micropolar continuum. In section 2, we review the multi-valued field theory and its path-dependency. In section 3, we consider the non-integrability conditions of the micropolar continuum from a viewpoint of the multi-valued field theory. In section 4, we introduce a line element and connection structure of the micropolar continuum. The geometric objects are expressed as the elastic and plastic parts. Then, the elastic and plastic parts of non-integrability conditions and the topological singularities of micropolar continuum are related to the non-Riemannian geometric objects. In section 5, the path-dependency due to the topological singularities in the micropolar continuum is described by the geometric objects. In section 6, we discuss an application of the geometric approach to an elasto-plastic mechanics such as an energy release rate around a crack tip and a constrained deformation of crystal. Moreover, based on a continuum theory of defects, a relationship between the non-locality of micropolar continuum and a kind of disclination called a kink band is considered. Then, a new expression of kink band is derived from a deformation gradient tensor which includes a disclination. The final section is the conclusion. Throughout of this paper, we use the Einstein summation convention.

2. Review of multi-valued field theory and path-dependency

In this section, we briefly review a theory of multi-valued field. Let us first define an $n$ -dimensional differentiable manifold $M$ with a coordinate system $(x^{a})$ . A coordinate transformation from $(x^{a})$ to $(x^{i})$ is expressed by Kleinert [12 –14]:

x^{a} \to x^{i} + ξ^{i} (x^{j}),

(1)

where $ξ^{i} = ξ^{i} (x^{j})$ is an infinitesimal vector. Then, the following commutation relations are considered [12 –14, 59, 60]:

V_{jk}^{i} = [\frac{\partial}{\partial x^{j}}, \frac{\partial}{\partial x^{k}}] ξ^{i},

(2)

W_{jkh}^{i} = [\frac{\partial}{\partial x^{j}}, \frac{\partial}{\partial x^{k}}] \frac{\partial ξ^{i}}{\partial x^{h}},

(3)

where the bracket $[]$ is a simple product defined as follows, and it is not a Lie bracket of a vector field:

[\frac{\partial}{\partial x^{j}}, \frac{\partial}{\partial x^{k}}] \equiv \frac{\partial}{\partial x^{j}} \frac{\partial}{\partial x^{k}} - \frac{\partial}{\partial x^{k}} \frac{\partial}{\partial x^{j}} .

(4)

If the vectors $ξ^{i}$ and $\partial ξ^{i} / \partial x^{h}$ are single-valued, $V_{jk}^{i}$ and $W_{jkh}^{i}$ satisfy the integrability conditions $V_{jk}^{i} = 0$ and $W_{jkh}^{i} = 0$ , respectively. If the vectors $ξ^{i}$ and $\partial ξ^{i} / \partial x^{h}$ are multi-valued, $V_{jk}^{i}$ and $W_{jkh}^{i}$ satisfy $V_{jk}^{i} \neq 0$ and $W_{jkh}^{i} \neq 0$ , respectively. The non-integrability conditions $V_{jk}^{i} \neq 0$ and $W_{jkh}^{i} \neq 0$ mean that $ξ^{i}$ and $\partial ξ^{i} / \partial x^{h}$ do not satisfy the equality of mixed partial derivatives of the Clairaut–Schwarz–Young theorem [61 –64]. Moreover, as mentioned in Scheidegger [65], a derivative of $ξ^{i}$ is related to the Cauchy–Riemann equations [66, 67].

The integrability condition also is related to the Poincaré lemma [68 –71]. For a one-form:

{\tilde{ξ}}^{i} \equiv \frac{\partial ξ^{i}}{\partial x^{j}} d x^{j},

(5)

the non-integrability condition $V_{jk}^{i} \neq 0$ is expressed by $d {\tilde{ξ}}^{i} \neq 0$ . In this case, the Poincaré lemma for de Rham cohomology [68, 72 –74] does not hold for the multi-valued function. Then, the property of multi-valued function is influenced on the topology of reference domain. Especially, a topological charge $b^{i}$ characterizes the integrability of $V_{jk}^{i}$ [2]:

b^{i} = \oint_{C} d ξ^{i} = \frac{1}{2} \int_{S} V_{jk}^{i} d S^{jk},

(6)

where $d S^{jk} = d x^{j} \land d x^{k}$ , $C$ is a closed curve, and $S$ is an oriented smooth surface. The integration path $C$ in equation (6) is defined not on the whole manifold $M$ but on a local region around the topological charge. When $ξ^{i}$ is multi-valued, a domain of $ξ^{i}$ is not simply connected from the Poincaré lemma or the de Rham cohomology. Therefore, by the Poincaré lemma, the multi-valued fields are defined in the non-contractible domain. Then, the multi-valued function $ξ^{i}$ has a path-dependency. Moreover, the topological charge is geometrically represented by a “discrepancy” along a closed curve $C$ [27], which is called non-evanescible circuit [65].

The continuity condition of $V_{jk}^{i}$ is given by (e.g., [2]):

\frac{\partial V_{kl}^{i}}{\partial x^{j}} + \frac{\partial V_{lj}^{i}}{\partial x^{k}} + \frac{\partial V_{jk}^{i}}{\partial x^{l}} = 0 .

(7)

$W_{jkh}^{i}$ is also related to the continuity conditions (e.g., [12, 14, 75]):

\frac{\partial W_{klh}^{i}}{\partial x^{j}} + \frac{\partial W_{ljh}^{i}}{\partial x^{k}} + \frac{\partial W_{jkh}^{i}}{\partial x^{l}} = 0 .

(8)

When the condition (7) holds, equation (2) follows from $\partial ξ^{i} / \partial x^{j}$ [16, 23].

3. Micropolar continuum with elastic and plastic deformations and integrability conditions of multi-valued fields

In this section, we consider a linear elasticity such as small displacements and rotations for a micropolar continuum. Field equations and basic quantities are first reviewed. Then, incompatibility conditions are considered for plastic and elastic deformations. After that, a relation between the integrability conditions and the incompatibility conditions are discussed.

3.1. Field equations and deformation quantities

In the following, we consider an isotropic micropolar continuum as a three-dimensional differentiable manifold, and let $x^{i}, (i = 1, 2, 3)$ be the local coordinate system based on Eringen [51, 52] and Lazar and Maugin [55]. The displacements, the stresses, and the couple stress are defined on this body. Then, the basic equations in the microscopic continuum are given by the field equations, the constitutive relations, the kinematic relations, the compatibility equations, and the stress boundary conditions.

The equations of motion of the micropolar continuum are expressed by:

\frac{\partial σ_{j}^{i}}{\partial x^{j}} = ρ \frac{d^{2} u^{i}}{d t^{2}},

(9)

\frac{\partial μ_{j}^{i}}{\partial x^{j}} + ε_{jk}^{i} σ_{k}^{j} = ρ I \frac{d^{2} ϕ^{i}}{d t^{2}},

(10)

where $σ_{j}^{i}$ is a stress tensor, $μ_{j}^{i}$ is a couple stress tensor, $u^{i}$ is a displacement vector, and $ϕ^{i}$ is a microrotation vector. $ρ$ is a density of material, $I$ is a microinertia moment, and $ε_{jk}^{i}$ is the Levi-Civita tensor. Moreover, a bend–twist tensor is given by the microrotation vector [51, 52, 55]:

\frac{\partial ϕ^{i}}{\partial x^{j}} \equiv κ_{j}^{i} .

(11)

The kinematic relation of the rotation vector is given by:

ω^{i} = \frac{1}{2} ε_{jk}^{i} \frac{\partial u^{k}}{\partial x^{j}},

(12)

where $ω^{i}$ is the macroscopic rotation and is different from the microrotation $ϕ^{i}$ . For the kinematic relation of the strain, a microstrain tensor $γ_{j}^{i}$ is defined by a gradient of displacement $\partial u^{i} / \partial x^{j}$ and the microrotation vector $ϕ^{i}$ [51, 55, 76]:

γ_{j}^{i} \equiv \frac{\partial u^{i}}{\partial x^{j}} + ε_{jk}^{i} ϕ^{k} .

(13)

The strain tensor given by the displacement is defined by:

e_{j}^{i} = \frac{1}{2} (\frac{\partial u^{i}}{\partial x^{j}} + \frac{\partial u^{j}}{\partial x^{i}}) .

(14)

The constitutive relations of the stress tensor and the couple stress tensor for the isotropic continuum are expressed by Eringen [52]:

σ_{j}^{i} = λ δ_{j}^{i} e_{k}^{k} + (2 μ + η) e_{j}^{i} - η ω_{j}^{i} - η ε_{jk}^{i} ϕ^{k},

(15)

μ_{j}^{i} = α δ_{j}^{i} κ_{k}^{k} + β κ_{j}^{i} + γ κ_{i}^{j},

(16)

where $λ$ and $μ$ are the Léme constant, and the other coefficients $α, β, γ,$ and $η$ are determined by the microscopic structure of the continuum.

The traction boundary conditions in macroscopic and microscopic balance law are given by Clayton et al. [41]:

t^{a} = Σ^{ab} n_{b}, ({\bar{t}}_{σ})^{\bar{α}} = σ^{\bar{α} b} n_{b}, ({\bar{t}}_{μ})^{\bar{α}} = μ^{\bar{α} b} n_{b} .

(17)

The gradient of displacement is related to the deformation gradient tensor through the description of deformations on the relation between position $x^{i}$ at time $t = t$ and position ${\bar{x}}^{i}$ at time $t = 0$ [77]:

\frac{\partial {\bar{x}}^{i}}{\partial x^{j}} = δ_{j}^{i} - \frac{\partial u^{i}}{\partial x^{j}},

(18)

where $δ_{j}^{i}$ is the Kronecker delta.

In the linear couple stress theory where the microrotation vector is equivalent to the macroscopic rotation, there are 15 unknown variables for the displacement $u^{i}$ , the macroscopic rotation $ω^{i}$ , the strain $γ_{j}^{i}$ , the stress tensor $σ_{j}^{i}$ , and the stress couple tensor $μ_{j}^{i}$ in the two-dimensional case. On the contrary, there are also 15 equations for the field equations, the constitutive relations, and the kinematic relations constructed by these variables. Therefore, these unknown variables can be obtained by giving three boundary conditions (e.g., [78]).

3.2. Incompatibility conditions of micropolar continuum

If the deformation tensors $γ_{j}^{i}$ and $κ_{j}^{i}$ are incompatible, the total fields are decomposed into elastic and plastic parts independently [55]:

γ_{j}^{(T) i} \equiv \frac{\partial u^{i}}{\partial x^{j}} + ε_{jk}^{i} ϕ^{k} = γ_{j}^{(E) i} + γ_{j}^{(P) i},

(19)

κ_{j}^{(T) i} \equiv \frac{\partial ϕ^{i}}{\partial x^{j}} = κ_{j}^{(E) i} + κ_{j}^{(P) i} .

(20)

In the following, the total field, the elastic contribution, and the plastic contribution are denoted by superscripts $(T)$ , $(E)$ , and $(P)$ , respectively. The elastic part of deformation tensors is related to the stress tensor and couple stress tensor through an elastic potential $Ψ (γ_{j}^{(E) i}, κ_{j}^{(E) i})$ [79]:

σ_{j}^{i} = \frac{\partial Ψ}{\partial γ_{j}^{(E) i}}, μ_{j}^{i} = \frac{\partial Ψ}{\partial κ_{j}^{(E) i}} .

(21)

Moreover, a plastic potential $f (σ_{j}^{i}, μ_{j}^{i})$ gives a relationship between a velocity of plastic part of microstrain tensor ${\overset{\cdot}{γ}}_{j}^{(P) i}$ and bend–twist tensor ${\overset{\cdot}{κ}}_{j}^{(P) i}$ [79]:

{\overset{\cdot}{γ}}_{j}^{(P) i} = Λ \frac{\partial f}{\partial σ_{j}^{i}}, {\overset{\cdot}{κ}}_{j}^{(P) i} = Λ \frac{\partial f}{\partial μ_{j}^{i}},

(22)

where $Λ$ depends on the coordinates $x^{k}$ .

The total elasto-plastic fields are compatible and satisfy the following conditions:

ε_{j}^{kl} (\frac{\partial γ_{l}^{(T) i}}{\partial x^{k}} + ε_{km}^{i} κ_{l}^{(T) m}) = 0,

(23)

ε_{j}^{kl} \frac{\partial κ_{l}^{(T) i}}{\partial x^{k}} = 0 .

(24)

When $κ_{l}^{(T) m} = 0$ , equation (23) is reduced to the Saint-Venant strain compatibility conditions [80]: $Inc \overset{\leftrightarrow}{ε}$ , where $Inc$ denotes the incompatibility operator, and $\overset{\leftrightarrow}{ε}$ is compatible with a strain tensor. However, each elastic field and plastic field are incompatible. For the elastic fields, the incompatibility equations are given by:

α_{j}^{i} = ε_{j}^{kl} (\frac{\partial γ_{l}^{(E) i}}{\partial x^{k}} + ε_{km}^{i} κ_{l}^{(E) m}),

(25)

θ_{j}^{i} = ε_{j}^{kl} \frac{\partial κ_{l}^{(E) i}}{\partial x^{k}} .

(26)

Equations (25) and (26) are related to the first Bianchi identity and the second Bianchi identity, respectively [55, 81]:

\partial α_{j}^{i} \partial x^{j} = 0, \partial θ_{j}^{i} \partial x^{j} = 0 .

(27)

Equation (27) represents the conservation law for dislocation line and the always closure for disclination line, respectively.

The incompatibility equations for the plastic fields are given by:

α_{j}^{i} = - ε_{j}^{kl} (\frac{\partial γ_{l}^{(P) i}}{\partial x^{k}} + ε_{km}^{i} κ_{l}^{(P) m}),

(28)

θ_{j}^{i} = - ε_{j}^{kl} \frac{\partial κ_{l}^{(P) i}}{\partial x^{k}} .

(29)

The above incompatibility quantities $α_{j}^{i}$ and $θ_{j}^{i}$ are considered as the dislocation density tensor and the disclination density tensor, respectively. It has been shown that there are two types of dislocation density tensors—one is obtained from the dislocation density tensor, and the other is equations (26) or (29) obtained from the bend–twist tensor [82]. These disclination density tensors differ from each other when there exists an incompatibility of distortion tensor. In other words, an incompatibility in a strain space is given by the difference between the two disclination density tensors.

Using the incompatibility equations, the characteristic vectors are defined. The general Burgers vector $B^{i}$ and the rotation vector $Ω^{i}$ called the Frank vector are associated with the Burgers surface $S$ [56, 57]:

B^{i} = \int_{S} (α_{j}^{i} - ε_{kl}^{i} θ_{j}^{k} x^{l}) d S^{j},

(30)

Ω^{i} = \int_{S} θ_{j}^{i} d S^{j} .

(31)

In equation (30), the Burgers vector involves the material point coordinates $x^{l}$ and is not translation invariant.

3.3. Incompatibility conditions of micropolar continuum and integrability conditions of multi-valued field

In the micropolar continuum, let us consider a coordinate transformation of the displacement and the microrotation fields:

x^{a} \to X^{i} = x^{i} + u^{i} (x^{j}),

(32)

y^{a} \to Y^{i} = y^{i} + ϕ^{i} (x^{j}) .

(33)

Then, we define the following commutation relations:

P_{jk}^{i} = [\frac{\partial}{\partial x^{j}}, \frac{\partial}{\partial x^{k}}] u^{i},

(34)

Q_{jk}^{i} = [\frac{\partial}{\partial x^{j}}, \frac{\partial}{\partial x^{k}}] ϕ^{i} .

(35)

According to equation (2), $P_{jk}^{i} \neq 0$ and $Q_{jk}^{i} \neq 0$ hold when $u^{i}$ and $ϕ^{i}$ are multi-valued, respectively.

As mentioned in the previous section, the Cauchy–Riemann equations are related to the derivative of $u^{i}$ and $ϕ^{i}$ . This means that $P_{jk}^{i} = 0$ and $Q_{jk}^{i} = 0$ hold when the derivatives of $u^{i}$ and $ϕ^{i}$ are differentiable and the second-order derivatives are continuous. On the contrary, $P_{jk}^{i} \neq 0$ and $Q_{jk}^{i} \neq 0$ hold when the gradient of displacement in equation (13) and the bend–twist tensor (11) may not be differentiable in the sense that the derivative does not exists, or the second-order derivatives are discontinuous. In this case, $P_{jk}^{i} \neq 0$ and $Q_{jk}^{i} \neq 0$ hold if the derivatives of $u^{i}$ and $ϕ^{i}$ are even single-valued. Then, the gradient of displacement and the microrotation induces a singularity as kink (existence condition of kink).

The integrability conditions (34) and (35) are also rewritten in terms of deformation tensors in the total fields. From equations (13) and (11), the compatibility condition (23) is expressed by:

ε_{j}^{kl} P_{kl}^{i} = 0 .

(36)

From equation (11), the compatibility condition (24) is expressed by:

ε_{j}^{kl} Q_{kl}^{i} = 0 .

(37)

Therefore, the compatibility of total field is equivalent to the integrability of displacement and microrotation in multi-valued field. Based on this relation, we denote the integrability quantities $P_{jk}^{i}$ and $Q_{jk}^{i}$ as $P_{jk}^{(T) i}$ and $Q_{jk}^{(T) i}$ . Then, similar to the deformation tensors, the integrability quantities $P_{jk}^{(T) i}$ and $Q_{jk}^{(T) i}$ are decomposed into the elastic part and the plastic part:

P_{jk}^{(T) i} = P_{jk}^{(E) i} + P_{jk}^{(P) i}, Q_{jk}^{(T) i} = Q_{jk}^{(E) i} + Q_{jk}^{(P) i} .

(38)

From equations (19) and (20), each parts are given by:

P_{jk}^{(E) i} = \frac{\partial γ_{k}^{(E) i}}{\partial x^{j}} - \frac{\partial γ_{j}^{(E) i}}{\partial x^{k}} - (ε_{kl}^{i} κ_{j}^{(E) l} - ε_{jl}^{i} κ_{k}^{(E) l}),

(39)

P_{jk}^{(P) i} = \frac{\partial γ_{k}^{(P) i}}{\partial x^{j}} - \frac{\partial γ_{j}^{(P) i}}{\partial x^{k}} - (ε_{kl}^{i} κ_{j}^{(P) l} - ε_{jl}^{i} κ_{k}^{(P) l}),

(40)

Q_{jk}^{(E) i} = \frac{\partial κ_{k}^{(E) i}}{\partial x^{j}} - \frac{\partial κ_{j}^{(E) i}}{\partial x^{k}},

(41)

Q_{jk}^{(P) i} = \frac{\partial κ_{k}^{(P) i}}{\partial x^{j}} - \frac{\partial κ_{j}^{(P) i}}{\partial x^{k}} .

(42)

Then, multiplying the Levi-Civita tensor on both sides of equation (38), the elastic part of integrability quantity is given by an inverse sign of the plastic part of integrability quantity:

P_{jk}^{(E) i} = - P_{jk}^{(P) i}, Q_{jk}^{(E) i} = - Q_{jk}^{(P) i} .

(43)

Equation (43) indicates an interaction between the elastic and plastic parts. Moreover, using equations (39) and (41), the incompatibility quantities $α_{j}^{i}$ and $θ_{j}^{i}$ are expressed by the elastic integrability quantities:

α_{j}^{i} = \frac{1}{2} ε_{j}^{kl} P_{kl}^{(E) i}, θ_{j}^{i} = \frac{1}{2} ε_{j}^{kl} Q_{kl}^{(E) i} .

(44)

From equations (43) and (44), the same incompatibility quantities are given by the plastic integrability quantities:

α_{j}^{i} = - \frac{1}{2} ε_{j}^{kl} P_{kl}^{(P) i}, θ_{j}^{i} = - \frac{1}{2} ε_{j}^{kl} Q_{kl}^{(P) i} .

(45)

Since the Burgers vector and Frank vector are described by the incompatibility, the integrability quantities are related to these characteristic vectors:

\begin{matrix} B^{i} = 2 \int_{S} ε_{j}^{kl} (P_{kl}^{(E) i} - ε_{pq}^{i} Q_{kl}^{(E) p} x^{q}) d S^{j} \\ = - 2 \int_{S} ε_{j}^{kl} (P_{kl}^{(P) i} - ε_{pq}^{i} Q_{kl}^{(P) p} x^{q}) d S^{j}, \end{matrix}

(46)

\begin{matrix} Ω^{i} = 2 \int_{S} ε_{j}^{kl} Q_{kl}^{(E) i} d S^{j} \\ = - 2 \int_{S} ε_{j}^{kl} Q_{kl}^{(P) i} d S^{j} . \end{matrix}

(47)

Thus, the incompatibility in micropolar continuum including the plastic part is expressed by the topological charge of multi-valued field. In this case, when the displacement and microrotation vectors are multi-valued, a domain of these vectors is not simply connected from the Poincaré lemma or the de Rham cohomology. Especially, the Burgers vector $B^{i}$ includes both the microstrain tensor $γ_{j}^{i}$ and the microrotation vector $ϕ^{i}$ . In case of the classical elasticity $ϕ^{i} = 0$ , if a double curl vanishes for a symmetric strain tensor, the strain tensor satisfies the integrability condition [12]. This means that the double curl is related to the integrability condition.

4. Geometric interpretation of multi-valued fields in micropolar continuum with plastic part

In this section, we show geometrically how the incompatibility conditions in micropolar continuum is related to the integrability conditions in multi-valued field based on a higher-order vector bundle.

4.1. Connection structures and geometric objects in second-order vector bundle

The micropolar continuum is regarded as a non-local continuum because of the microrotation vector. The term “non-local” means that the continuum has a hierarchical structure on a scale ranging from macro to micro and does not represent the globality over the entire continuum. Such “non-locality,”“fluctuation,” or “anisotropy” has been discussed by a property of Finsler geometry and higher-order geometry [30 –34, 50, 83 –85]. In the classical elasticity, a macroscopic displacement vector is attached to each point of manifold, and then the geometric framework is given by a first-order vector bundle [27]. In the micropolar continuum, since the microrotation vectors are considered as more microscopic level, the geometric structure of the micropolar continuum is described in more higher-order space, i.e., a second-order vector bundle [28]. However, this previous study discusses only the geometric features of the elastic component of the micropolar continuum. As shown in the previous section, a theory of micropolar continuum can be described by the elastic and plastic contributions. Therefore, in this section, we generalize the result in the previous study [28] into elasto-plastic micropolar continuum.

Let us first define $(x^{i}) = (x^{1}, x^{2}, x^{3})$ as local coordinates in a three-dimensional micropolar continuum including plastic part. In the micropolar continuum, since two types of vectors, displacement $u^{i}$ and microrotation $ϕ^{i}$ , give defects, we consider a second-order vector bundle with these vectors as local coordinates. Then, we introduce a line element or element of support which is local coordinates of the bundle over the body and put the line element as $(x^{i}, u^{i}, ϕ^{i})$ . Since we consider the line element for the second-order vector bundle, the connection structure is given by the connection coefficients $Γ_{jk}^{(T) i}$ , $Θ_{jk}^{(T) i}$ , and $Σ_{jk}^{(T) i}$ corresponding to each variable:

D X^{i} = d X^{i} + Γ_{jk}^{(T) i} X^{j} d x^{k} + Θ_{jk}^{(T) i} X^{j} d u^{k} + Σ_{jk}^{(T) i} X^{j} d ϕ^{k},

(48)

where $X^{i}$ is an arbitrary vector. $Γ_{jk}^{i}$ , $Θ_{jk}^{i}$ , and $Σ_{jk}^{i}$ are the connection coefficients. It is remarked that the notation $(T)$ in the connection coefficients denotes the total field.

Based on the concept of intrinsic parallelism in the Finsler geometry [30 –33], covariant derivatives of the displacement vector and the microrotation vector in the total field are given by another connection coefficients:

\begin{matrix} δ u^{i} = d u^{i} + A_{jk}^{(T) i} u^{j} d x^{k} + C_{jk}^{(T) i} u^{j} d ϕ^{k} \\ = d u^{i} + {\tilde{A}}_{k}^{(T) i} d x^{k} + {\tilde{C}}_{k}^{(T) i} d ϕ^{k}, \end{matrix}

(49)

\begin{matrix} δ ϕ^{i} = d ϕ^{i} + D_{jk}^{(T) i} ϕ^{j} d x^{k} + E_{jk}^{(T) i} ϕ^{j} d u^{k} \\ = d ϕ^{i} + {\tilde{D}}_{k}^{(T) i} d x^{k} + {\tilde{E}}_{k}^{(T) i} d u^{k}, \end{matrix}

(50)

where we put:

{\tilde{A}}_{k}^{(T) i} = A_{jk}^{(T) i} u^{j}, {\tilde{C}}_{k}^{(T) i} = C_{jk}^{(T) i} u^{j},

(51)

{\tilde{D}}_{k}^{(T) i} = D_{jk}^{(T) i} ϕ^{j}, {\tilde{E}}_{k}^{(T) i} = E_{jk}^{(T) i} ϕ^{j} .

(52)

From definition of the bend–twist (11), $κ_{j}^{(T) i}$ is given by the derivative of microrotation vector and is not related to the displacement vector. Therefore, the displacement does not influence on the covariant derivative $δ ϕ^{i}$ and, we can put $E_{jk}^{(T) i} = 0$ and ${\tilde{E}}_{j}^{(T) i} = 0$ .

As mentioned in the previous section, since the total field of micropolar continuum is decomposed into the elastic and plastic parts, we assume that the connection coefficients are also divided into these two parts:

Γ_{jk}^{(T) i} = Γ_{jk}^{(E) i} + Γ_{jk}^{(P) i}, Θ_{jk}^{(T) i} = Θ_{jk}^{(E) i} + Θ_{jk}^{(P) i},

(53)

Σ_{jk}^{(T) i} = Σ_{jk}^{(E) i} + Σ_{jk}^{(P) i}, {\tilde{A}}_{k}^{(T) i} = {\tilde{A}}_{k}^{(E) i} + {\tilde{A}}_{k}^{(P) i},

(54)

{\tilde{C}}_{k}^{(T) i} = {\tilde{C}}_{k}^{(E) i} + {\tilde{C}}_{k}^{(P) i}, {\tilde{D}}_{k}^{(T) i} = {\tilde{D}}_{k}^{(E) i} + {\tilde{D}}_{k}^{(P) i},

(55)

where the notations $(E)$ and $(P)$ in the connection coefficients denote the elastic part and the plastic part, respectively.

From equations (49) and (50), $d u^{i}$ and $d ϕ^{i}$ are expressed by:

d u^{k} = δ u^{k} - {\tilde{A}}_{l}^{(T) k} d x^{l} - {\tilde{C}}_{l}^{(T) k} d ϕ^{l},

(56)

d ϕ^{k} = δ ϕ^{k} - {\tilde{D}}_{l}^{(T) k} d x^{l},

(57)

where we use the assumption ${\tilde{E}}_{l}^{(T) k} = 0$ . Then, substituting equations (56) and (57) into the covariant derivative of total field (48), we have:

\begin{matrix} D X^{i} = d X^{i} + (Γ_{jk}^{(T) i} - Θ_{jl}^{(T) i} {\tilde{A}}_{k}^{(T) l} - Σ_{jl}^{(T) i} {\tilde{D}}_{k}^{(T) l}) X^{j} d x^{k} \\ - Θ_{jl}^{(T) i} {\tilde{C}}_{k}^{(T) l} X^{j} d ϕ^{k} + Θ_{jk}^{(T) i} X^{j} δ u^{k} + Σ_{jk}^{(T) i} X^{j} + δ ϕ^{k} . \end{matrix}

(58)

Once again substituting equation (57) into $d ϕ^{l}$ in equation (58), we get the following form:

D X^{i} = d X^{i} + F_{jk}^{(T) i} X^{j} d x^{k} + Θ_{jk}^{(T) i} X^{j} δ u^{k} + H_{jk}^{(T) i} X^{j} δ ϕ^{k},

(59)

where the connection coefficients are defined by:

F_{jk}^{(T) i} = Γ_{jk}^{(T) i} - Θ_{jl}^{(T) i} N_{k}^{(T) l} - Σ_{jl}^{(T) i} {\tilde{D}}_{k}^{(T) l},

(60)

H_{jk}^{(T) i} = Σ_{jk}^{(T) i} - Θ_{jl}^{(T) i} {\tilde{C}}_{k}^{(T) l} .

(61)

M_{k}^{(T) l} = {\tilde{C}}_{h}^{(T) l} {\tilde{D}}_{k}^{(T) h},

(62)

N_{k}^{(T) l} = {\tilde{A}}_{k}^{(T) l} - M_{k}^{(T) l} .

(63)

Since the total field is independently decomposed into the elastic and plastic contributions, we assume that the connection coefficient $M_{k}^{(T) l}$ is also decomposed as follows:

M_{k}^{(T) l} = {\tilde{C}}_{h}^{(E) l} {\tilde{D}}_{k}^{(E) h} + {\tilde{C}}_{h}^{(P) l} {\tilde{D}}_{k}^{(P) h} .

(64)

Then, the connection coefficient $N_{k}^{(T) l}$ is expressed by the elastic and plastic parts:

\begin{matrix} N_{k}^{(T) l} = N_{k}^{(E) l} + N_{k}^{(P) l} \\ = ({\tilde{A}}_{k}^{(E) l} - {\tilde{M}}_{k}^{(E) l}) + ({\tilde{A}}_{k}^{(P) l} - {\tilde{M}}_{k}^{(P) l}) . \end{matrix}

(65)

The decompositions of other connection coefficients are given by:

\begin{matrix} F_{jk}^{(T) i} = (Γ_{jk}^{(E) i} - Θ_{jl}^{(E) i} N_{k}^{(E) l} - Σ_{jl}^{(E) i} {\tilde{D}}_{k}^{(E) l}) \\ + (Γ_{jk}^{(P) i} - Θ_{jl}^{(P) i} N_{k}^{(P) l} - Σ_{jl}^{(P) i} {\tilde{D}}_{k}^{(P) l}) \\ = F_{jk}^{(E) i} + F_{jk}^{(P) i}, \end{matrix}

(66)

\begin{matrix} H_{jk}^{(T) i} = (Σ_{jk}^{(E) i} - Θ_{jl}^{(E) i} {\tilde{C}}_{k}^{(E) l}) + (Σ_{jk}^{(P) i} - Θ_{jl}^{(P) i} {\tilde{C}}_{k}^{(P) l}) \\ = H_{jk}^{(E) i} + H_{jk}^{(P) i}, \end{matrix}

(67)

where we use the independency between the elastic part of connection coefficient and the plastic part of connection coefficient.

An adapted frame for the line element $(x^{i}, u^{i}, ϕ^{i})$ is given by $(d x^{i}, δ u^{i}, δ ϕ^{i})$ . In this case, we introduce the derivatives $X_{| j}^{i}$ , $X^{i} |_{j}$ , and $X^{i} | |_{j}$ defined by:

\begin{matrix} X_{| j}^{i} = \frac{δ X^{i}}{δ x^{j}} + F_{jk}^{(T) i} X^{k} \\ = \frac{δ X^{i}}{δ x^{j}} + F_{jk}^{(E) i} X^{k} + F_{jk}^{(P) i} X^{k}, \end{matrix}

(68)

\begin{matrix} X^{i} |_{j} = \frac{\partial X^{i}}{\partial u^{j}} + Θ_{jk}^{(T) i} X^{k} \\ = \frac{\partial X^{i}}{\partial u^{j}} + Θ_{jk}^{(E) i} X^{k} + Θ_{jk}^{(P) i} X^{k}, \end{matrix}

(69)

\begin{matrix} X^{i} | |_{j} = \frac{δ X^{i}}{δ ϕ^{j}} + H_{jk}^{(T) i} X^{k} \\ = \frac{δ X^{i}}{δ ϕ^{j}} + H_{jk}^{(E) i} X^{k} + H_{jk}^{(P) i} X^{k}, \end{matrix}

(70)

where we put the derivatives as follows:

\begin{matrix} \frac{δ}{δ x^{j}} = \frac{\partial}{\partial x^{j}} - N_{j}^{(T) k} \frac{\partial}{\partial u^{k}} - {\tilde{D}}_{j}^{(T) k} \frac{\partial}{\partial ϕ^{k}} \\ = \frac{\partial}{\partial x^{j}} - N_{j}^{(E) k} \frac{\partial}{\partial u^{k}} - N_{j}^{(P) k} \frac{\partial}{\partial u^{k}} \\ - {\tilde{D}}_{j}^{(E) k} \frac{\partial}{\partial ϕ^{k}} - {\tilde{D}}_{j}^{(P) k} \frac{\partial}{\partial ϕ^{k}}, \end{matrix}

(71)

\begin{matrix} \frac{δ}{δ ϕ^{j}} = \frac{\partial}{\partial ϕ^{j}} - {\tilde{C}}_{j}^{(T) k} \frac{\partial}{\partial u^{k}} \\ = \frac{\partial}{\partial ϕ^{j}} - {\tilde{C}}_{j}^{(E) k} \frac{\partial}{\partial u^{k}} - {\tilde{C}}_{j}^{(P) k} \frac{\partial}{\partial u^{k}} . \end{matrix}

(72)

Then, using equations (68)–(70), the connection structure (59) can be rewritten by:

D X^{i} = X_{| j}^{i} d x^{j} + X^{i} |_{j} δ u^{j} + X^{i} | |_{j} δ ϕ^{j} .

(73)

4.2. Intrinsic parallelism of displacement and microrotation

The covariant derivatives $δ u^{i}$ and $δ ϕ^{i}$ describe intrinsic behaviors of displacement vector $u^{i}$ and microrotation vector $ϕ^{i}$ , respectively. Especially, these vectors take a specific orientation with excited state when $δ u^{i} = 0$ and $δ ϕ^{i} = 0$ hold [30 –34]. In the following, we show that the geometric objects of multi-valued field in micropolar continuum is given by the covariant derivatives $δ u^{i}$ and $δ ϕ^{i}$ .

Let us first consider the state given by the covariant derivatives $δ u^{i} = 0$ and $δ ϕ^{i} = 0$ . Then, the following relations are obtained from equations (49) and (50):

d u^{i} + ({\tilde{A}}_{k}^{(E) i} + {\tilde{A}}_{k}^{(P) i}) d x^{k} + ({\tilde{C}}_{k}^{(E) i} + {\tilde{C}}_{k}^{(P) i}) d ϕ^{k} = 0,

(74)

d ϕ^{i} + ({\tilde{D}}_{k}^{(E) i} + {\tilde{D}}_{k}^{(P) i}) d x^{k} = 0 .

(75)

Moreover, from equations (65) and (75), equation (74) can be rewritten by:

d u^{i} + ({\tilde{N}}_{k}^{(E) i} + N_{k}^{(P) i}) d x^{k} = 0 .

(76)

Using equations (75) and (76), the connection structure (59) under the intrinsic conditions $δ u^{i} = 0$ and $δ ϕ^{i} = 0$ is given by the connection coefficient $F_{jk}^{(E) i}$ and $F_{jk}^{(P) i}$ only:

D X^{i} = d X^{i} + F_{jk}^{(E) i} X^{j} d x^{k} + F_{jk}^{(P) i} X^{j} d x^{k} .

(77)

Next, let us consider curvature tensors and torsion tensors which can be obtained from the Ricci identity (e.g., [46]). Especially, the torsion tensor represents a non-Riemannian property [47, 86]. In the total field, a curvature tensor $R_{jkh}^{(T) i}$ and torsion tensors $T_{jk}^{(T) i}$ and $R_{(u) jk}^{(T) i}$ and torsion tensor or curvature of nonlinear connection $R_{(ϕ) jk}^{(T) i}$ are given by:

X_{| j | k}^{i} - X_{| k | j}^{i} = - R_{(u) jk}^{(T) l} \frac{\partial X^{i}}{\partial u^{l}} - R_{(ϕ) jk}^{(T) l} \frac{\partial X^{i}}{\partial ϕ^{l}} + T_{jk}^{(T) l} X_{| l}^{i} + R_{jkh}^{(T) i} X^{h},

(78)

where the geometric objects are given by as follows:

R_{(u) jk}^{(T) l} = \frac{δ N_{j}^{(T) l}}{δ x^{k}} - \frac{δ N_{k}^{(T) l}}{δ x^{j}},

(79)

R_{(ϕ) jk}^{(T) l} = \frac{δ {\tilde{D}}_{j}^{(T) l}}{δ x^{k}} - \frac{δ {\tilde{D}}_{k}^{(T) l}}{δ x^{j}},

(80)

T_{jk}^{(T) l} = F_{jk}^{(T) l} - F_{kj}^{(T) l},

(81)

R_{jkh}^{(T) i} = \frac{δ F_{jh}^{(T) i}}{δ x^{k}} - \frac{δ F_{kh}^{(T) i}}{δ x^{j}} + F_{kl}^{(T) i} F_{jh}^{(T) l} - F_{jl}^{(T) i} F_{kh}^{(T) l} .

(82)

The torsion tensors or curvature of nonlinear connection $R_{(u) jk}^{(T) l}$ and $R_{(ϕ) jk}^{(T) l}$ describes a discrepancy of the gradient of displacement vector and the gradient of microrotation vector, respectively.

Since the total field is independently decomposed into the elastic part and the plastic part, the geometric objects are also decomposed into the following manner:

\begin{matrix} R_{(u) jk}^{(T) l} = \frac{δ N_{j}^{(E) l}}{δ x^{k}} - \frac{δ N_{k}^{(E) l}}{δ x^{j}} + \frac{δ N_{j}^{(P) l}}{δ x^{k}} - \frac{δ N_{k}^{(P) l}}{δ x^{j}} \\ = R_{(u) jk}^{(E) l} + R_{(u) jk}^{(P) l}, \end{matrix}

(83)

\begin{matrix} R_{(ϕ) jk}^{(T) l} = \frac{δ {\tilde{D}}_{j}^{(E) l}}{δ x^{k}} - \frac{δ {\tilde{D}}_{k}^{(E) l}}{δ x^{j}} + \frac{δ {\tilde{D}}_{j}^{(P) l}}{δ x^{k}} - \frac{δ {\tilde{D}}_{k}^{(P) l}}{δ x^{j}} \\ = R_{(ϕ) jk}^{(E) l} + R_{(ϕ) jk}^{(P) l}, \end{matrix}

(84)

\begin{matrix} T_{jk}^{(T) l} = F_{jk}^{(E) l} - F_{kj}^{(E) l} + F_{jk}^{(P) l} - F_{kj}^{(P) l} \\ = T_{jk}^{(E) l} + T_{jk}^{(P) l}, \end{matrix}

(85)

\begin{matrix} R_{jkh}^{(T) i} = \frac{δ F_{jh}^{(E) i}}{δ x^{k}} - \frac{δ F_{kh}^{(E) i}}{δ x^{j}} + F_{kl}^{(E) i} F_{jh}^{(E) l} - F_{jl}^{(E) i} F_{kh}^{(E) l} \\ + \frac{δ F_{jh}^{(P) i}}{δ x^{k}} - \frac{δ F_{kh}^{(P) i}}{δ x^{j}} + F_{kl}^{(P) i} F_{jh}^{(P) l} - F_{jl}^{(P) i} F_{kh}^{(P) l} \\ = R_{jkh}^{(E) i} + R_{jkh}^{(P) i} . \end{matrix}

(86)

The connection coefficient $F_{jk}^{(E) i}$ is given by the connection coefficients ${\tilde{A}}_{j}^{(E) i}$ and ${\tilde{D}}_{j}^{(E) i}$ . These connections represent the interaction among $(x)$ -field, $(u)$ -field, and $(ϕ)$ -field of elastic part. This means that the curvature tensor $R_{jkh}^{(E) i}$ shows a geometric structure of unified elastic field in micropolar continuum. In a similar way, the curvature tensor $R_{jkh}^{(P) i}$ expresses a geometric structure of unified plastic field in micropolar continuum. In this case, the curvature tensors are related to the Euler–Schouten curvature tensor $S_{jk}^{(E) α}$ and $S_{jk}^{(P) α}$ which are relative curvatures in regard to the enveloping space [87, 88]:

R_{ijkh}^{(E)} = \frac{1}{2} (S_{ik}^{(E) α} S_{jh}^{(E) α} - S_{ih}^{(E) α} S_{jk}^{(E) α}),

(87)

R_{ijkh}^{(P)} = \frac{1}{2} (S_{ik}^{(P) α} S_{jh}^{(P) α} - S_{ih}^{(P) α} S_{jk}^{(P) α}),

(88)

where the Greek index $α$ means components of tensors in the unified field of $(u)$ -field and $(ϕ)$ -field. Here, the covariant expressions of curvature tensors, $R_{ijkh}^{(E)}$ and $R_{ijkh}^{(P)}$ , are given by the contraction of metric tensor. In this case, the curvature tensor embedded in the unified space also has a meaning as a Gauss–Codazzi relation. From a viewpoint of continuum mechanics, the existence of curvature tensor is related to a disclination quadrupole structure in kink band or hardening mechanism of kink band [89, 90].

4.3. Relationships between integrability conditions and geometric objects

In this section, the quantities of integrability conditions (38) are expressed by the torsion tensor or the curvature of nonlinear connection. When the intrinsic parallelism of displacement and microrotation vectors holds, relations (75) and (76) are rewritten by:

\frac{\partial ϕ^{i}}{\partial x^{k}} = - ({\tilde{D}}_{k}^{(E) i} + {\tilde{D}}_{k}^{(P) i}),

(89)

\begin{matrix} \frac{\partial u^{i}}{\partial x^{k}} = - (N_{k}^{(E) i} + N_{k}^{(P) i}) \\ = - ({\tilde{A}}_{k}^{(E) i} - M_{k}^{(E) i}) - ({\tilde{A}}_{k}^{(P) i} - M_{k}^{(P) i}) . \end{matrix}

(90)

From equations (89) and (90), $u^{i}$ and $ϕ^{i}$ are the functions of $x^{i}$ only under the conditions $δ u^{i} = 0$ and $δ ϕ^{i} = 0$ . Geometrically, if $u^{i}$ and $ϕ^{i}$ are the functions of $x^{i}$ , the Finsler field is reduced to a non-Riemannian space. These reduction processes are known as the osculating conditions from the Finsler space (e.g., [83 –85, 91 –93]). Then, the coefficients ${\tilde{D}}_{k}^{(E) i}$ and ${\tilde{D}}_{k}^{(P) i}$ play a role of nonlinear connections which express an interaction between $(x)$ -field and $(ϕ)$ -field. On the contrary, the coefficients $N_{k}^{(E) i}$ and $N_{k}^{(P) i}$ are nonlinear connections which express an interaction between $(x)$ -field and $(u)$ -field. From equations (64) and (65), these nonlinear connections $N_{k}^{(E) i}$ and $N_{k}^{(P) i}$ are expressed by ${\tilde{C}}_{k}^{(E) i}, {\tilde{C}}_{k}^{(P) i}$ , ${\tilde{D}}_{k}^{(E) i}$ , and ${\tilde{D}}_{k}^{(P) i}$ which give the interactions between $(x)$ -field and $(ϕ)$ -field. Therefore, the nonlinear connections $N_{k}^{(E) i}$ and $N_{k}^{(P) i}$ describe an influence of microrotation in elastic and plastic fields. As a concrete example, the components of the nonlinear connection are described in Appendix 1 when there is a dislocation line along the $x^{3}$ -axis.

Let us consider geometric expressions of integrability conditions from a viewpoint of second-order vector bundle. From equations (89) and (90), the microstrain tensor (19) and bend–twist tensor (20) are given by the connection coefficients:

γ_{k}^{(E) i} = - {\tilde{A}}_{k}^{(E) i}, γ_{k}^{(P) i} = - {\tilde{A}}_{k}^{(P) i},

(91)

ε_{kl}^{i} ϕ^{l} = - (M_{k}^{(E) i} + M_{k}^{(P) i}) = - M_{k}^{(T) i},

(92)

κ_{k}^{(E) i} = - {\tilde{D}}_{k}^{(E) i}, κ_{k}^{(P) i} = - {\tilde{D}}_{k}^{(P) i} .

(93)

Then, the elastic and plastic parts of integrability quantities (39)–(42) are rewritten by:

P_{jk}^{(E) i} = \frac{\partial {\tilde{A}}_{j}^{(E) i}}{\partial x^{k}} - \frac{\partial {\tilde{A}}_{k}^{(E) i}}{\partial x^{j}} - (ε_{jl}^{i} {\tilde{D}}_{k}^{(E) l} - ε_{kl}^{i} {\tilde{D}}_{j}^{(E) l}),

(94)

P_{jk}^{(P) i} = \frac{\partial {\tilde{A}}_{j}^{(P) i}}{\partial x^{k}} - \frac{\partial {\tilde{A}}_{k}^{(P) i}}{\partial x^{j}} - (ε_{jl}^{i} {\tilde{D}}_{k}^{(P) l} - ε_{kl}^{i} {\tilde{D}}_{j}^{(P) l}),

(95)

Q_{jk}^{(E) i} = \frac{\partial {\tilde{D}}_{j}^{(E) i}}{\partial x^{k}} - \frac{\partial {\tilde{D}}_{k}^{(E) i}}{\partial x^{j}},

(96)

Q_{jk}^{(P) i} = \frac{\partial {\tilde{D}}_{j}^{(P) i}}{\partial x^{k}} - \frac{\partial {\tilde{D}}_{k}^{(P) i}}{\partial x^{j}} .

(97)

Moreover, since we assume the osculating conditions $δ u^{i} = 0$ and $δ ϕ^{i} = 0$ , the connection coefficients in equations (89) and (90) are the functions of $x^{i}$ . Then, the integrability quantities of displacement and microrotation are equivalent to the torsion tensors or the curvatures of nonlinear connection:

P_{jk}^{(E) i} = R_{(u) jk}^{(E) i}, P_{jk}^{(P) i} = R_{(u) jk}^{(P) i},

(98)

Q_{jk}^{(E) i} = R_{(ϕ) jk}^{(E) i}, Q_{jk}^{(P) i} = R_{(ϕ) jk}^{(P) i} .

(99)

Therefore, the integrability quantities of the displacement vector $u^{i}$ and the microrotation vector $ϕ^{i}$ are geometrically related to the torsion tensors or the curvatures of nonlinear connection in the non-Riemannian space. Moreover, from equations (44) and (45), the incompatibility quantities are expressed by the torsion tensor or the curvature of nonlinear connection:

α_{j}^{i} = \frac{1}{2} ε_{j}^{kl} R_{(u) kl}^{(E) i} = - \frac{1}{2} ε_{j}^{kl} R_{(u) kl}^{(P) i},

(100)

θ_{j}^{i} = \frac{1}{2} ε_{j}^{kl} R_{(ϕ) kl}^{(E) i} = - \frac{1}{2} ε_{j}^{kl} R_{(ϕ) kl}^{(P) i} .

(101)

5. Geometric interpretations of characteristic vectors for micropolar continuum

In this section, we discuss the path-dependence of displacement field and microrotation field based on the geometric objects. When an integral of a multi-valued function along a closed curve surrounding the singularity is considered, the value of the integral differs depending on the path due to the multi-valence of the integrand.

Let us first consider the path-dependency of Burgers vector $B^{i}$ for the displacement. From equation (46), $B^{i}$ consists of the integrability quantities $P_{jk}^{(E) i}$ and $Q_{jk}^{(E) i}$ or $P_{jk}^{(P) i}$ and $Q_{jk}^{(P) i}$ . From equations (94)–(97), these quantities are expressed by the connection coefficients ${\tilde{A}}_{j}^{(E) i}$ and ${\tilde{D}}_{j}^{(E) i}$ or ${\tilde{A}}_{j}^{(P) i}$ and ${\tilde{D}}_{j}^{(P) i}$ . The connection coefficient ${\tilde{A}}_{k}^{(E) i}$ represents the interaction between $(x)$ -field and $(u)$ -field. The ${\tilde{D}}_{j}^{(E) i}$ represents the interaction between $(x)$ -field and $(ϕ)$ -field. Therefore, the path dependency of Burgers vector is geometrically characterized by the interaction among $(x)$ -field, $(u)$ -field, and $(ϕ)$ -field:

B^{i} = 2 \int_{S} ε_{j}^{kl} {\frac{\partial {\tilde{A}}_{k}^{(E) i}}{\partial x^{l}} - \frac{\partial {\tilde{A}}_{l}^{(E) i}}{\partial x^{k}} - (ε_{kh}^{i} {\tilde{D}}_{l}^{(E) h} - ε_{lh}^{i} {\tilde{D}}_{k}^{(E) h}) - ε_{pq}^{i} (\frac{\partial {\tilde{D}}_{k}^{(E) p}}{\partial x^{l}} - \frac{\partial {\tilde{D}}_{l}^{(E) p}}{\partial x^{k}}) x^{q}} d S^{j} .

(102)

On the contrary, the path-dependency of Frank vector $Ω^{i}$ for the microrotation is related to the integrability quantity $Q_{jk}^{(E) i}$ or $Q_{jk}^{(P) i}$ . These quantities are given by the nonlinear connection ${\tilde{D}}_{j}^{(E) i}$ or ${\tilde{D}}_{j}^{(P) i}$ from the previous subsection. Therefore, $Ω^{i}$ is geometrically expressed by the interaction between $(x)$ -field and $(ϕ)$ -field:

Ω^{i} = 2 \int_{S} ε_{j}^{kl} (\frac{\partial {\tilde{D}}_{k}^{(E) i}}{\partial x^{l}} - \frac{\partial {\tilde{D}}_{l}^{(E) i}}{\partial x^{k}}) d S^{j} .

(103)

Moreover, from equations (44) and (45), the integrability quantities $P_{jk}^{(E) i}$ and $Q_{jk}^{(E) i}$ or $P_{jk}^{(P) i}$ and $Q_{jk}^{(P) i}$ are related to the torsion tensors or the curvature of nonlinear connection $R_{(u) jk}^{(E) i}$ and $R_{(ϕ) jk}^{(E) i}$ or $R_{(u) jk}^{(P) i}$ and $R_{(ϕ) jk}^{(P) i}$ . This means that the path-dependency of micropolar continuum is related to the torsion tensors or the curvatures of nonlinear connection in the non-Riemannian space.

6. Discussion

In this study, a relationship between the deformation quantities of micropolar continuum and geometric objects is considered. Here, we discuss a geometric interpretation of the energy release rate around a crack tip, the Taylor–Bishop–Hill theory and the mathematical rotation as the contents related to the micropolar continuum. Moreover, a formation of kink is discussed from a viewpoint of surface dislocation [94].

6.1. Energy release rate and geometric object

As mentioned in the previous sections, the incompatibility tensor is geometrically related to the torsion tensor or the curvature of nonlinear connection. In fracture mechanics, the incompatibility tensor gives an energy release rate $G_{i}$ around the crack tip in the elasto-plastic body [95]:

G_{i} = J_{i}^{(P)} + J_{i}^{(G)},

(104)

where $J_{i}^{(P)}$ is the Peach–Koehler force acting on the dislocation and the disclination:

J_{i}^{(P)} = \int_{D} σ^{jk} ε_{jil} α_{k}^{l} dD + \int_{D} μ^{jk} ε_{jil} θ_{k}^{l} dD .

(105)

In equation (105), $D$ is a domain around the crack tip, and $J_{i}^{(G)}$ is a generalized force acting on the singular stress field due to the geometric shape of the crack tip. $J_{i}^{(G)}$ is given by the energy momentum tensor $U_{i}^{j}$ :

J_{i}^{(G)} = \int_{C} U_{i}^{j} n_{j} dS,

(106)

where $C$ is a closed curve surrounding the crack, and $dS$ is a surface element of the region $D$ . Moreover, $n_{j}$ is a unit normal vector on the surface of $D$ . The energy momentum tensor $U_{i}^{j}$ is defined using the strain energy density $W$ of the micropolar continuum [96]:

U_{i}^{j} = W δ_{i}^{j} - (σ^{jk} \frac{\partial u^{k}}{\partial x^{i}} - μ^{jk} \frac{\partial ϕ^{k}}{\partial x^{i}}) .

(107)

From equations (100) and (101), the Peach–Koehler force is expressed by the torsion tensor or the curvature of nonlinear connection:

\begin{matrix} J_{i}^{(P)} = \frac{1}{2} \int_{D} ε_{jil} ε_{k}^{rs} σ^{jk} R_{(u) rs}^{(E) l} dD + \frac{1}{2} \int_{D} ε_{jil} ε_{k}^{rs} μ^{jk} R_{(ϕ) rs}^{(E) l} dD \\ = - \frac{1}{2} \int_{D} ε_{jil} ε_{k}^{rs} σ^{jk} R_{(u) rs}^{(P) l} dD - \frac{1}{2} \int_{D} ε_{jil} ε_{k}^{rs} μ^{jk} R_{(ϕ) rs}^{(P) l} dD . \end{matrix}

(108)

Moreover, the dislocation density flux tensor $I_{j}^{i}$ is given by the displacement incompatible tensor, and the rotation density flux tensor $Ω_{j}^{i}$ is given by the microrotation incompatible tensor. This means that these fluxes are also characterized by the torsion tensor or the curvature of nonlinear connection:

I_{j}^{i} = \frac{1}{2} ε_{kl}^{i} ε_{j}^{rs} R_{(u) rs}^{(E) l} ν^{k} = - \frac{1}{2} ε_{kl}^{i} ε_{j}^{rs} R_{(u) rs}^{(P) l} ν^{k},

(109)

Ω_{j}^{i} = \frac{1}{2} ε_{kl}^{i} ε_{j}^{rs} R_{(ϕ) rs}^{(E) l} ν^{k} = - \frac{1}{2} ε_{kl}^{i} ε_{j}^{rs} R_{(ϕ) rs}^{(P) l} ν^{k},

(110)

where $ν^{k}$ is a velocity of disclination line. The Peach–Koehler force represents the force acting on the stress field of the defect distributed at the crack tip in the material fracture process. Therefore, expression (108), in which the Peach–Koehler force is given by the torsion or the curvature of the nonlinear connection, means that the fracture phenomenon of the micropolar continuum can be discussed in terms of geometric quantities on non-Riemannian spaces.

6.2. Deformation of constrained crystal and geometric object

The deformation quantities of a continuum including the microstructure is related to the plastic deformation of the constrained crystal. When a single crystal is deformed under tension and compression, a moment is generated due to the bias applied to the loading axis. In this case, the relationship among the lattice rotation, continuum spin, and plastic spin is given by the Hosford mathematical rule [97, 98]. If a lattice rotation has the same sign as the bias moment, it is stable and the Schmid rule is considered to be correct. If not, the constraint from the surrounding matrix or tool will prevent the enlargement of bias, and the local stresses will cause partly non-uniform deformations (kink or secondary slip) accompanied by some unstable lattice rotations. When the Schmid rule holds, the plastic deformation of crystals is described by constrained deformation theories such as Taylor–Bishop–Hill theory [76, 99, 100]. According to this theory, the constrained deformation $d {\tilde{u}}^{i}$ of a crystal is given by the deformation quantities due to intercrystalline slip $d {\tilde{u}}^{{s} i}$ and the additional rotation $d {\tilde{u}}^{{r} i}$ :

d {\tilde{u}}^{i} = d {\tilde{u}}^{{s} i} + d {\tilde{u}}^{{r} i},

(111)

where $d {\tilde{u}}^{{s} i}$ is defined by a slip tensor $τ_{j}^{i}$ :

d {\tilde{u}}^{{s} i} = τ_{j}^{i} d {\tilde{x}}^{j} .

(112)

Moreover, based on the discussion of micromorphic continuum [101], we assume that $d {\tilde{u}}^{{r} i}$ is given by the microrotation $ϕ_{j}^{i} \equiv ε_{jk}^{i} ϕ^{k}$ :

d {\tilde{u}}^{{r} i} = ϕ_{j}^{i} d {\tilde{x}}^{j} .

(113)

In Taylor–Bishop–Hill theory, let $s^{(A) i}$ be the slip vector in a specific slip system $A$ , and let $n_{j}^{(A)}$ be the unit normal vector perpendicular to the slip plane. Then, the slip tensor is given by:

τ_{j}^{i} = \sum_{A} τ^{(A)} s^{(A) i} n_{j}^{(A)} .

(114)

On the contrary, comparing the relationship among the deformation amounts of the micropolar continuum (13), (112), and (113), $d {\tilde{u}}^{{s} i}$ due to the intercrystalline slip and $d {\tilde{u}}^{{r} i}$ due to the additional rotation correspond to the displacement gradient $\partial u^{i} / \partial x^{j}$ and the microrotation $ϕ_{j}^{i}$ , respectively. The displacement gradient is expressed by the nonlinear connection $N_{j}^{(T) i}$ from equations (65) and (90). The microrotation $ϕ_{j}^{i}$ is described by another nonlinear connection $M_{j}^{(T) i}$ from equation (92). From the above relations, the intercrystalline slip $d {\tilde{u}}^{{s} i}$ and additional rotation $d {\tilde{u}}^{{r} i}$ under the constrained deformation of crystals are geometrically characterized by nonlinear connections $N_{j}^{(T) i}$ and $M_{j}^{(T) i}$ .

6.3. Non-locality of micropolar continuum and connection of kink band

In the micromechanics, the disclination as the topological charges are given by the microrotation. An example of such defects is the kink found in metallic materials. Bullough and Bilby [94] have derived the theory of plane surface dislocations required for the analysis of martensite crystallography. When the matrix of martensite is undeformed, a relationship [102] between the Burgers vector and the deformation gradient is expressed by:

\begin{matrix} B^{i} = B_{j}^{i} w^{j} \\ = ε_{jkl} v^{k} [E_{l}^{(+) i} - E_{l}^{(-) i}] ε_{jmn} v^{m} p^{n} \\ = (E_{l}^{(+) i} - δ_{l}^{i}) p^{l} . \end{matrix}

(115)

The detailed notations in equation (115) are denoted in Appendix 2. Expression (115) is equivalent to the equation on the rank-1 connection between the kink band and the undeformed matrix for the formation of the kink without fractures along kink boundary (e.g., [103]) and the equation on 0-lattice theory for the geometry of grain and phase boundaries [104]. Ball and Carstensen [105], Iwaniec et al. [106] and Ball [107] regard this rank-1 connection as the Hadamard [108] jump compatibility.

$E_{j}^{i}$ and $D_{j}^{i}$ have a reciprocal relationship, and $D_{j}^{i}$ is a deformation gradient tensor as mentioned in equation (18). Regarding the deformation tensor, Bullough and Bilby [94] were submitted at a time when the concept of disclination was still in its infancy. In this paper, the deformation gradient tensor including disclinations was obtained. Therefore, let us apply this deformation gradient tensor including disclinations to the equation on the surface dislocation given by Bullough and Bilby [94], a new equation on surface dislocations with disclination can be obtained as (see Appendix 2):

B_{j}^{i} = ε_{jkl} v^{k} [U_{l}^{(+) i} - U_{l}^{(-) i}] - ε_{jkl} v^{k} [χ_{l}^{(+) i} - χ_{l}^{(-) i}],

(116)

which can describe the kink interface including the disclination. Using the dislocation content $B_{j}^{i}$ , the Burgers vector with dislocations and disclinations can be obtained: $B^{i} = B_{j}^{i} w^{j}$ . If there are no internal rotations $ϕ^{i} = 0$ , expression (116) reduces to the Burgers vector (115) with dislocations only. Inamura [103] recently discovered that when a shear band crosses a kink boundary (interface), a disclination pair must exist at the kink boundary (interface) to maintain the rank-1 connection without fractures.

For the formation of the kink, the rank-1 connection between the kink band and the undeformed matrix is given by a shear deformation and a rigid rotation [103]. Especially, when two kink bands satisfy the rank-1 connection, a rotation angle of disclination or rank-1 connection $ω^{*}$ is given by angles of a rigid rotation $θ^{1}$ and $θ^{2}$ : $ω^{*} = θ^{1} - θ^{2}$ . On the contrary, from a viewpoint of the micropolar continuum, the Frank vector as a characteristic vector of disclination (??) is expressed by the microrotation vector $ϕ^{i}$ :

Ω^{i} = \int_{C} κ_{j}^{i} d x^{j} = \int_{C} d ϕ^{i} = Δ ϕ^{i} \neq 0,

(117)

where $C$ is the Burgers circuit, and $ϕ^{i}$ is regarded as a multi-valued. The integration path $C$ in equation (117) is defined not in the entire micropolar continuum but in a local region around the defect. Therefore, the rotation angle of disclination $ω^{*}$ due to the connection of the kink bands is expressed as the Frank vector $Ω^{i}$ in the micropolar continuum. In this case, the angles of rotation in the kink bands $θ^{1}$ and $θ^{2}$ correspond to the change in microrotation vector $Δ ϕ^{i}$ . Moreover, the Frank vector is also related to the incompatibility obtained from the bend–twist $κ_{j}^{i}$ using the differential form [82]. The bend–twist $κ_{j}^{i}$ in equation (117) appears when a non-localized micropolar continuum including microscopic rotations is considered. The relation (117) shows that the non-local quantity of such microscopic continuum is related to the kink band. Therefore, the formation of the kink band is characterized by the non-locality of the microrotation vector in the micropolar continuum, based on the continuum theory of defects.

7. Conclusion

In this paper, geometric structures of micropolar continuum with elastic and plastic deformations are discussed. The integrability conditions for the elastic and plastic fields are obtained for the macro displacement and microrotation vectors. These incompatibility conditions are expressed by the integrabilities of multi-valued macro displacement and microrotation vectors. In this case, the Burgers vector and the Frank vector are expressed by the elastic or the plastic part of integrability conditions.

From a viewpoint of differential geometry, the deformation of micropolar continuum can be described by a connection structure of second-order vector bundle. The line element of second-order vector bundle is defined by the macro displacement and microrotation vectors. Then, the geometric objects are decomposed into the elastic part and the plastic part independently. Especially, when the intrinsic parallelisms of macro displacement and microrotation vectors hold, the micropolar continuum can be discussed in non-Riemannian space. In this case, the torsion tensor or the curvature of nonlinear connection is related to the integrability conditions of elastic field and plastic field. This means that the path-dependency of the Burgers and Frank vectors is given by the non-Riemannian properties of multi-valued field. From the above, the geometric structure of the micropolar continuum including the plastic field can be described by the geometric quantity on the second-order vector bundle. Moreover, using the relationship between the incompatibility condition and the geometric objects, the energy release rate around a crack tip, the Taylor–Bishop–Hill theory, and the mathematical rotation are discussed. Based on a continuum theory of defects, the non-locality of micropolar continuum is related to a kink band.

Footnotes

Appendix 1

Appendix 2 Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by JSPS KAKENHI (grant nos JP19H05117 and JP21H00090).

ORCID iD

Takahiro Yajima

References

Umezawa

Matsumoto

Tachiki

Thermo field dynamics and condensed states. Amsterdam: North-Holl and Publishing Company, 1982.

Wadati

Quantum field theory of crystals and extended objects. Phys Rep 1979; 50(2): 87–155.

Katanaev

MO.

Wedge dislocation in the geometric theory of defects. Theor Math Phys 2003; 135: 733–744.

Katanaev

MO.

Geometric theory of defects. Phys Usp 48: 675–701.

Katanaev

Volovich

IV.

Theory of defects in solids and three-dimensional gravity. Ann Phys 1992; 216(1): 1–28.

Kondo

. Geometry of elastic deformation and incompatibility. In: Kondo

(ed.) Memoirs of the unifying study of the basic problems in engineering sciences by means of geometry. Vol. I. Tokyo, Japan: Gakujutsu Bunken Fukyu-Kai, 1955, pp. 361–373.

Kröner

Kontinuumstheorie der Versetzungen und Eigenspannungen. Berlin: Springer, 1958.

Kröner

. Continuum theory of defects. In: Balian

(ed.) Physics of defects (Proc. Les Houches XXXV). Amsterdam: North-Holland, 1981, pp. 217–315.

Yamasaki

Nagahama

Hodge duality and continuum theory of defects. J Phys A Math Gen 1999; 32(44): L475–L481.

10.

Yamasaki

Nagahama

A deformed medium including a defect field and differential forms. J Phys A Math Gen 2002; 35(16): 3767–3778.

11.

Yamasaki

Nagahama

Energy integral in fracture mechanics (J-integral) and Gauss–Bonnet theorem. Z Angew Math Mech 2008; 88(6): 515–520.

12.

Kleinert

Gauge fields in condensed matter. Vol. 2. Singapore: World Scientific Publishing, 1989.

13.

Kleinert

Nonholonomic mapping principle for classical and quantum mechanics in spaces with curvature and torsion. Gen Relat Gravit 2000; 32: 769–839.

14.

Kleinert

Multivalued fields in condensed matter, electromagnetism, and gravitation. Singapore: World Scientific Publishing, 2008.

15.

Blasone

Jizba

Vitiello

Quantum field theory and its macroscopic manifestations: boson condensation, ordered patterns, and topological defects. Singapore: World Scientific Publishing, 2011.

16.

Wadati

Matsumoto

Umezawa

Extended objects in crystals. Phys Rev B 1978; 18(8): 4077–4095.

17.

Wadati

Matsumoto

Umezawa

General theory of grain boundaries in crystal. Solid State Commun 1978; 27(9): 923–925.

18.

Wadati

Matsumoto

Umezawa

A new derivation of surface waves in crystal. Solid State Commun 1978; 27(9): 927–929.

19.

Wadati

Takahashi

Umezawa

A self-consistent field theory of crystals. Phys Lett A 1977; 62(4): 255–257.

20.

Wadati

Matsumoto

Takahashi

, et al. General theory of dislocation. Phys Lett A 1977; 62(4): 258–260.

21.

Matumoto

Papastamatiou

Umezawa

The boson transformation and the vortex solutions. Nucl Phys B 1975; 97(1): 90–124.

22.

Matumoto

Sodano

Umezawa

Extended objects in quantum systems and soliton solutions. Phys Rev D 1979; 19(2): 511–516.

23.

Wadati

Matsumoto

Umezawa

Bags in relativistic quantum field theory with spontaneously broken symmetry. Phys Rev D 1978; 18(4): 1192–1195.

24.

Freeman

Vitiello

Dissipative neurodynamics in perception forms cortical patterns that are stabilized by vortices. J Phys Conf Ser 2009; 174: 012011.

25.

Freeman

Vitiello

Vortices in the brain waves. Int J Mod Phys B 2010; 24(17): 3269–3295.

26.

Freeman

Livi

Obinata

, et al. Cortical phase transitions, non-equilibrium thermodynamics and the time-dependent Ginzburg–Landau equation. Int J Mod Phys B 2012; 26(6): 1250035.

27.

Yajima

Nagahama

Finsler geometry of topological singularities for multi-valued fields: applications to continuum theory of defects. Ann Phys 2016; 528(11–12): 845–851.

28.

Yajima

Nagahama

Connection structures of topological singularity in micromechanics from a viewpoint of generalized Finsler space. Ann Phys 2020; 532(12): 2000306.

29.

Zubov

LM.

Nonlinear theory of dislocations and disclinations in elastic bodies. Berlin: Springer, 1997.

30.

Ikeda

On the covariant differential of spin direction in the Finslerian deformation theory of ferromagnetic substances. J Math Phys 1981; 22(12): 2831–2834.

31.

Ikeda

Some structurological features induced by the internal variable in the theory of fields in Finsler spaces. Prog Theor Phys 1981; 65(6): 2075–2078.

32.

Ikeda

On the intrinsic behavior of the internal variable in the Finslerian gravitational field. J Math Phys 1985; 26(5): 958–960.

33.

Ikeda

A Finslerian extension of the gravitational field—III. Ind J Pure Appl Math 1990; 21(9): 841–845.

34.

Stavrinos

Ikeda

Connection considerations of gravitational field in Finsler spaces. Int J Theor Phys 2006; 45(4): 743–749.

35.

Clayton

JD.

On Finsler geometry and applications in mechanics: review and new perspectives. Adv Math Phys 2015; 2015: 828475.

36.

Clayton

JD.

Finsler-geometric continuum mechanics and the micromechanics of fracture in crystals. J Micromech Mol Phys 2016; 1(3n4): 1640003.

37.

Clayton

JD.

Finsler geometry of nonlinear elastic solids with internal structure. J Geom Phys 2017; 112: 118–146.

38.

Clayton

JD.

Finsler-geometric continuum dynamics and shock compression. Int J Fract 2017; 208: 53–78.

39.

Clayton

JD.

Generalized Finsler geometric continuum physics with applications in fracture and phase transformations. Z Angew Math Phys 2017; 68: 9.

40.

Clayton

JD.

Generalized pseudo-Finsler geometry applied to the nonlinear mechanics of torsion of crystalline solids. Int J Geom Method Mod Phys 2018; 15(7): 1850113.

41.

Clayton

McDowell

Bammann

DJ.

Modeling dislocations and disclinations with finite micropolar elastoplasticity. Int J Plast 2006; 22: 210–256.

42.

Clayton

JD.

Mesoscale models of interface mechanics in crystalline solids: a review. J Mater Sci 2018; 53: 5515–5545.

43.

Abreu

Durand

Thomas

The asymmetric seismic moment tensor in micropolar media. Bull Seismol Soc Am 2018; 108(3A): 1160–1170.

44.

Abreu

Durand

Understanding micropolar theory in the earth sciences II: the seismic moment tensor. Pure Appl Geophys 2021; 178, 4325–4343.

45.

Abreu

Durand

Understanding micropolar theory in the earth sciences I: the eigenfrequency

ω_{r}

. Pure Appl Geophys 2022; 179, 915–932.

46.

Antonelli

Zastawniak

. Fundamentals of Finslerian diffusion with applications. In: Antonelli

(ed.) Handbook of Finsler geometry. Vol. 1. Dordrecht: Kluwer Academic Publishers, 2003, pp. 177–355.

47.

Nagahama

Teisseyre

Continuum theory of defects: advanced approaches. In: Teisseyre

Nagahama

Majewski

(eds) Physics of asymmetric continua: extreme and fracture processes. Berlin: Springer-Verlag, 2008, pp. 221–248.

48.

Nagahama

Teisseyre

Micromorphic continuum and fractal fracturing in the lithosphere. Pageoph 2000; 157: 559–574.

49.

Forest

Modeling slip, kink and shear banding in classical and generalized single crystal plasticity. Acta Mater 1998; 46(9): 3265–3281.

50.

Ikeda

Advanced studies in applied geometry. Sagamihara, Japan: Seizansha, 1995.

51.

Eringen

AC.

Linear theory of micropolar elasticity. J Math Mech 1966; 15(6): 909–923.

52.

Eringen

AC.

Microcontinuum field theories. New York: Springer-Verlag, 1999.

53.

Nowacki

Theory of micropolar elasticity. Wien: Springer-Verlag, 1972.

54.

Mindlin

RD.

Micro-structure in linear elasticity. Arch Ration Mech Anal 1964; 16: 51–78.

55.

Lazar

Maugin

GA.

On microcontinuum field theories: the Eshelby stress tensor and incompatibility conditions. Philos Mag 2007; 87(25): 3853–3870.

56.

deWit

. Theory of disclinations: II. Continuous and discrete disclinations in anisotropic elasticity. J Res Nat Bur Stand A Phys Chem 1973; 77A(1): 49–100.

57.

Kossecka

deWit

Disclination kinematics. Arc Mech 1997; 29(5): 633–651.

58.

Pommaret

J-F.

Parametrization of Cosserat equations. Acta Mech 2010; 215(1–4): 43–55.

59.

Kleinert

Gravity as a theory of defects in a crystal with only second gradient elasticity. Ann Phys 1987; 499(2): 117–119.

60.

Kleinert

Path integrals in quantum mechanics, statistics, and polymer physics, and financial markets. Singapore: World Scientific Publishing, 1995.

61.

Clairaut

Recherches générales sur le calcul intégral. HARS 1739 Mém 1741; 425–436.

62.

Clairaut

Sur l’intégration ou la construction des e

\overset{'}{q}

uations différentielles du premier ordre. HARS 1740, Mém 1742; 293–323.

63.

Schwarz

HA.

Communication. Arch Sci Phys Nat 1873; 48: 38–44.

64.

Young

WH.

IX.-On the conditions for the reversibility of the order of partial differentiation. Proc R Soc Edinb 1909; 29: 136–164.

65.

Scheidegger

AE.

Principles of geodynamics. Berlin: Springer-Verlag, 1982.

66.

Cauchy

AL.

Mémoire sur les intégrales définies. Mémoires présentés par divers savants á l’ Académie royale des sciences de l’Institut de France, 1 (read 22 August 1814, published 1827). Œuvres complètes d’Augustin Cauchy. Sér. 1 Tome 1, Paris: de l’Académie des sciences et sous les auspices de M. le ministre de l’Instruction publique, 1882, pp. 329–506.

67.

Riemann

Bernhard Riemann’s gesammelte mathematische Werke und wissenschaftlicher Nachlass (ed Weber

Dedekind

). Leipzig: B. G. Teubner, 1876.

68.

Nash

Sen

Topology and geometry for physicists. London: Academic Press, 1983.

69.

Nickerson

Spencer

Steenrod

NE.

Advanced calculus. Princeton, NJ: D. Van Nostrand, 1959.

70.

Poincaré

Sur les résidus des intégrales doubles. Acta Math 1887; 9: 321–380.

71.

Poincaré

Les méthodes nouvelles de la mécanique céleste. Vol. 3. Paris: Gauthier-Villars, 1899.

72.

de Rham

. Sur l’analysis situs des variétés à n dimensions. J Math Pures Appl 1931; 10(9): 115–200.

73.

de Rham

. Œuvres Mathématiques. Genève: L’Enseignement Mathématique, Université de Genève, 1981.

74.

Weil

Sur les théorèmes de de Rham. Comment Math Helv 1952; 26: 119–145.

75.

Cvetkovic

Nussinov

Zaanen

Topological kinematic constraints: dislocations and the glide principle. Philos Mag 2006; 86(20): 2995–3020.

76.

Kanatani

A note on crystal rotations under constrained deformation. J Japan Inst Met Mater 1980; 44(7): 836–837 (in Japanese).

77.

Yamasaki

Tensor analysis of dislocation–stress relationship based on the extended deformation gradient. Acta Geophys Pol 2005; 53(1): 1–12.

78.

Mindlin

RD.

Influence of couple-stresses on stress concentrations. Exp Mech 1963; 3: 1–7.

79.

Sekine

A Cosserat theory of crystal plasticity with application to analysis of lattice rotations under constrained deformation. J Japan Inst Met Mater 1977; 41(9): 874–882 (in Japanese).

80.

Baus

Lovett

Generalization of the stress tensor to nonuniform fluids and solids and its relation to the Saint-Venant’s strain compatibility conditions. Phys Rev Lett 1990; 65(14): 1781–1783.

81.

Hirano

Nagahama

Nonmetricity on Riemann–Cartan–Weyl manifold: its physical and mathematical meaning and application. Int J Geom Methods Mod Phys 2022; 19(10): 2250153.

82.

Yamasaki

Hasebe

Duality of the incompatibility tensor. Mater Trans 2020; 61(5): 875–877.

83.

Ikeda

On the theory of fields in Finsler spaces. J Math Phys 1981; 22(6): 1215–1218.

84.

Ikeda

Some physical aspects induced by the internal variable in the theory of fields in Finsler spaces. Nuovo Cim B 1981; 61(2): 220–228.

85.

Ikeda

Some structural features induced by the space-time metrical fluctuation in the theory of gravitational fields. Found Phys 1983; 13(6): 629–636.

86.

Ikeda

Review of applied geometry. Tokyo, Japan: Iwanami Shoten Book Center, 2015 (in Japanese).

87.

Kondo

Non-Riemannian and Finslerian approaches to the theory of yielding. Int J Enging Sci 1963; 1(1): 71–88.

88.

Schouten

JA.

Ricci-calculus, an introduction to tensor analysis and its geometrical applications. 2nd ed. Berlin: Springer-Verlag, 1954.

89.

Kimura

Ueta

Shizawa

Dislocation-based crystal plasticity FE analysis for kink band formation in Mg-based LPSO phase considering higher-order stress. Procedia Manuf 2018; 15: 1825–1832.

90.

Pertsev

Romanov

Vladimirov

VI.

Disclination–dislocation model for the kink bands in polymers and fibre composites. J Mat Sci 1981; 16: 2084–2090.

91.

Amari

. A theory of deformations and stresses of ferromagnetic substances by Finsler geometry. In: Kondo

(ed.) RAAG memoirs of the unifying study of basic problems in engineering and physical sciences by means of geometry. Vol. III. Tokyo, Japan: Gakujutsu Bunken Fukyu-Kai, 1962, pp. 193–214.

92.

Rund

The differential geometry of finsler spaces. Berlin: Springer-Verlag, 1959.

93.

Sakata

. A constructive approach to non-teleparallelism and non-metric representations of plastic material manifold by generalized diakoptical tearing II. Finslerian tearing. RAAG Res Notes 1970; 3–149: 1–27.

94.

Bullough

Bilby

BA.

Continuous distributions of dislocations: surface dislocations and the crystallography of martensitic transformations. Proc Phys Soc B 1956; 69: 1276–1286.

95.

Toriyama

The relationship between energy release rate and microstructures in Cosserat continuum. J Soc Mat Sci 1999; 48(10): 1155–1159 (in Japanese).

96.

Atkinson

Leppington

FG.

Some calculations of the energy-release rate G for cracks in micropolar and couple-stress elastic media. Int J Fract 1974; 10: 599–602.

97.

Hosford

WF.

On orientation changes accompanying slip and twinning. Text Cryst Solids 1977; 2: 175–182.

98.

Takahashi

Motohashi

Tsuchida

Development of plastic anisotropy in rolled aluminium sheets. Int J Plast 1996; 12(7): 935–949.

99.

Taylor

GI.

Plastic strain in metals. J Inst Metals 1938; 62: 307–324,

100.

Bishop

JFW

Hill

. XLVI. A theory of the plastic distortion of a polycrystalline aggregate under combined stresses. Philos Mag 1951; 42(327): 414–427.

101.

Muto

Kawada

Nagahama

. Micromorphic continuum with defects and Taylor–Bishop–Hill theory for polycrystals: anisotropic propagation of seismic waves and the Golebiewska gauge. In: Teisseyre

Takeo

Majewski

(eds) Earthquake source asymmetry, structural media and rotation effects. Berlin: Springer-Verlag, 2006: 317–328.

102.

Bilby

Bullough

Smith

Continuous distributions of dislocations: a new application of the methods of non-Riemannian geometry. Proc R Soc Lond A 1955; 231(1185): 263–273.

103.

Inamura

Geometry of kink microstructure analysed by rank-1 connection. Acta Mater 2019; 173: 270–280.

104.

Bollmann

. On the geometry of grain and phase boundaries, I. General theory. Philos Mag 1967; 16(140): 363–381.

105.

Ball

Carstensen

Compatibility conditions for microstructures and the austenite–martensite transition. Mat Sci Eng A 1999; 273–275: 231–236.

106.

Iwaniec

Verchota

Vogel

AL.

The failure of rank-one connections. Arch Ration Mech Anal 2002; 163: 125–169.

107.

Ball

JM.

Mathematical models of martensitic microstructure. Mat Sci Eng A 2004; 378: 61–69.

108.

Hadamard

Leçons sur la propagation des ondes et les équations de l’hydrodynamique, cours du Collège de France. Paris: Librairie Scientifique A. Hermann, 1903.